A History of Greek Mathematics and Astronomy

Page 31

The first to make systematic use of symbols in algebraical work was Diophantus of Alexandria (fl. about A. D. 250). He used (1) a sign for the unknown quantity, which he calls αριθμος {arithmos}, and compendia for its powers up to the sixth; (2) a sign ([Transcriber's Note: Symbol]) with the effect of our minus. The latter sign probably represents ΛΙ {LI}, an abbreviation for the root of the word λειπειν {leipein} (to be wanting); the sign for αριθμος {arithmos} ([Transcriber's Note: Symbol]) is most likely an abbreviation for the letters αρ {ar}; the compendia for the powers of the unknown are Δ^Υ {D^Y} for δυναμις {dynamis}, the square, Κ^Υ {K^Y} for κυβος {kybos}, the cube, and so on. Diophantus shows that he solved quadratic equations by rule, like Heron. His Arithmetica, of which six books only (out of thirteen) survive, contains a certain number of problems leading to simple equations, but is mostly devoted to indeterminate or semi-determinate analysis, mainly of the second degree. The collection is extraordinarily varied, and the devices resorted to are highly ingenious. The problems solved are such as the following (fractional as well as integral solutions being admitted): 'Given a number, to find three others such that the sum of the three, or of any pair of them, together with the given number is a square', 'To find four numbers such that the square of the sum plus or minus any one of the numbers is a square', 'To find three numbers such that the product of any two plus or minus the sum of the three is a square'. Diophantus assumes as known certain theorems about numbers which are the sums of two and three squares respectively, and other propositions in the Theory of Numbers. He also wrote a book On Polygonal Numbers of which only a fragment survives.

With Pappus and Diophantus the list of original writers on mathematics comes to an end. After them came the commentators whose names only can be mentioned here. Theon of Alexandria, the editor of Euclid, lived towards the end of the fourth century A. D. To the fifth and sixth centuries belong Proclus, Simplicius, and Eutocius, to whom we can never be grateful enough for the precious fragments which they have preserved from works now lost, and particularly the History of Geometry and the History of Astronomy by Aristotle's pupil Eudemus.

Such is the story of Greek mathematical science. If anything could enhance the marvel of it, it would be the consideration of the shortness of the time (about 350 years) within which the Greeks, starting from the very beginning, brought geometry to the point of performing operations equivalent to the integral calculus and, in the realm of astronomy, actually anticipated Copernicus.

Previous Page ||| First Page

T. L. Heath
A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath A History of Greek Mathematics and Astronomy

Learned Freeware

T. L. Heath
A History of Greek Mathematics and Astronomy